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Topology + Music

Topology + Music

Project Summary

This data expedition introduced students to “sliding windows and persistence” on time series data, which is an algorithm to turn one dimensional time series into a geometric curve in high dimensions, and to quantitatively analyze hybrid geometric/topological properties of the resulting curve such as “loopiness” and “wiggliness.”

Used the “Loop Ditty” software Chris Tralie created to visualize geometric curves that represent the music, projected down to 3D

Were able to quantitatively analyze where vocals occurred in music by looking at loops, and that classical music is “smoother” than rock music

Got a safe introduction to topological data analysis techniques for analyzing the curve in its true high dimensional embedding

Discovered trade-offs between analyzing data visually after projection and analyzing data with more abstract tools in high dimensions

In this expedition, musical audio data was the 1D time series in question, which is a fun and relatable way to explore these complicated time series analysis algorithms. Musical audio data is high dimensional (44100 samples per second), noisy, complex, and highly repetitive. It’s the repetitive nature and the global/local descriptions of music that have been of particular interest in my personal research on music information retrieval. As a result, my data expedition was more about exploring “complex data” as opposed to “big data.” Therefore, most of my efforts curating the data were directed at creating a web-based graphical user interface, which I called “Loop Ditty” (http://www.loopditty.net), which turned music into visual curves in a way that would allow students to visually explore and manipulate complex patterns in the sound that would be difficult to glean from the waveform alone. Figure 1 shows a screenshot from the LoopDitty software

In addition to visual exploration in 3D, students also used tools from topological data analysis to quantitatively analyze the musical curves in high dimensions. Their task was to make observations in both domains.

Marine mammals exhibit extreme physiological and behavioral adaptions that allow them to dive hundreds to thousands of meters underwater despite their need to breathe air at the surface. Through the development of new remote monitoring technologies, we are just beginning to understand the mechanisms by which they are able to execute these extreme behaviors. Long- term animal-borne tags can now record location, dive depth, and dive duration and then transmit these data to satellite receivers, enabling remote access to behavior occurring both many kilometers out to sea and several kilometers below the ocean surface.

The aim of this Data Expedition was for students to learn hands-on data visualization techniques using a variety of data types. Students first discussed how data visualization is useful, and tips to make graphs both visually appealing and easy to understand.